Vitaly Volpert

Elliptic Partial Differential Equations (eBook, PDF)

Volume 2: Reaction-Diffusion Equations

• Format: PDF

Produktbeschreibung

If we had to formulate in one sentence what this book is about, it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.

Produktdetails

• Verlag: Springer Basel
• Erscheinungstermin: 10. Mai 2014
• Englisch
• ISBN-13: 9783034808132
• Artikelnr.: 44225082

Autorenporträt

Vitaly Volpert started his scientific career in Russia and continued it in the USA and in France. He works on partial differential equations and on mathematical modelling in chemical physics, biology and medicine. He is an author of more than 200 scientific publications including three monographs.

Inhaltsangabe

I. Introduction to the theory of reaction-diffusion equations.- Chapter 1. Reaction-diffusion processes, models and applications.- Chapter 2. Methods of analysis.- Chapter 3. Reaction-diffusion problems in bounded domains.- Chapter 4. Reaction-diffusion problems on the whole axis.- II. Reaction-diffusion waves in cylinders.- Chapter 5. Monotone systems.- Chapter 6. Reaction-diffusion problems with convection.- Chapter 7. Reaction-diffusion systems with different diffusion coefficients.- Chapter 8. Nonlinear boundary conditions.- Chapter 9. Nonlocal reaction-diffusion equations.- Chapter 10. Multi-scale models in biology.- Bibliographical comments.- Concluding remarks.- Acknowledgements.- References.- Index.

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Chapter 1. Introduction.
Chapter 2. Function spaces and operators.
Chapter 3. A priori estimates.
Chapter 4. Normal solvability.
Chapter 5. Fredholm property.
Chapter 6. Formally adjoint problems.
Chapter 7. Elliptic problems with a parameter.
Chapter 8. Index of elliptic operators.
Chapter 9. Problems in cylinders.
Chapter 10. Non
Fredholm operators.
Chapter 11. Nonlinear Fredholm operators.
Supplement. Discrete operators.
Historical and bibliographical comments.
Acknowledgements.
References.

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I. Introduction to the theory of reaction-diffusion equations.- Chapter 1. Reaction-diffusion processes, models and applications.- Chapter 2. Methods of analysis.- Chapter 3. Reaction-diffusion problems in bounded domains.- Chapter 4. Reaction-diffusion problems on the whole axis.- II. Reaction-diffusion waves in cylinders.- Chapter 5. Monotone systems.- Chapter 6. Reaction-diffusion problems with convection.- Chapter 7. Reaction-diffusion systems with different diffusion coefficients.- Chapter 8. Nonlinear boundary conditions.- Chapter 9. Nonlocal reaction-diffusion equations.- Chapter 10. Multi-scale models in biology.- Bibliographical comments.- Concluding remarks.- Acknowledgements.- References.- Index.

Rezensionen

From the reviews:

"In the book under review the author concentrates on the Fredholm theory for (mainly linear) boundary value problems in unbounded domains. ... Both for a researcher in the field and for a mathematical research library, this is a must-have item." (Niels Jacob, The Mathematical Gazette, Vol. 98 (541), March, 2014)

"This very interesting book presents a systematic study of elliptic problems in unbounded domains of Rn. ... At the end of the book ... the author gives a comprehensive account of the large development of the elliptic theory from its birth in the XVIIIth century up to the present time. ... This historical account gives to the reader the full perspective on each topic as well as a complete and up-to-date bibliography. This is certainly an excellent starting point for any kind of further investigation." (Paolo Acquistapace, Mathematical Reviews, Issue 2012 g)

"The book is devoted to substantial and systematic presentation of the linear and nonlinear theory of elliptic partial differential equations in unbounded domains. ... The presentation of the results is mostly self-contained. Thus, the book can be recommended either for experts in the subject or for beginners." (Sergei V. Rogosin, Zentralblatt MATH, Vol. 1222, 2011)